# Calibration of utility function parameters controlling labor disutility

Consider the very basic utility function

$$u(c, n) = \log (c) - \alpha \frac{n^{1 + \frac{1}{\nu}}}{1 + \frac{1}{\nu}}$$

Where $c$ is consumption and $n$ is working hours. As the Frisch elasticity here is given by $\nu$, I would calibrate $\nu$ to be between 0.5 and 1.

However, I'm wondering what to do with $\alpha$: What do I calibrate that against?

• I have not seen a problem like this before so the reason I am ask for clarification is probably my ignorance: Are you asking if there should be any limits on $\alpha$ when you fit this utility function on some data using revealed preferences or are you asking something else? – Giskard Jun 1 '15 at 16:28
• @denesp I'm asking what to target at all. Is there a standard in the literature, is there a specific problem (revealed preferences) to target at - or do I have free hands at choosing the parameter to my liking? – FooBar Jun 1 '15 at 16:38
• I am sorry, I still don't understand. What are you trying to do, what is your goal? :) The only thing I see is that according to classical theory $\alpha > 0$ since working cuts into leisure time which is a useful good. It can well be that I am the only one confounded by your question, so feel free to ignore me and wait for someone else to answer. – Giskard Jun 1 '15 at 16:43
• Maybe @Foobar is asking about benchmarks about $\alpha$ from maybe experimental of econometric literature. – user157623 Jun 1 '15 at 18:19
• @user157623 That's pretty much it. I am aware of micro evidence for $\nu$, but have no idea what to target $\alpha$ against. – FooBar Jun 1 '15 at 18:21

$\alpha$ handles the conversion between the units in which labor is measured and the the units in which consumption is measured.

Consider the leisure part of the utility function: $$- \alpha \frac{n^{1 + \frac{1}{\nu}}}{1 + \frac{1}{\nu}}$$ We can rewrite it like this: $$- (\alpha^{\frac{1}{1 + \frac{1}{\nu}}})^{1 + \frac{1}{\nu}} \frac{n^{1 + \frac{1}{\nu}}}{1 + \frac{1}{\nu}} = - \frac{(n\cdot (\alpha^{\frac{1}{1 + \frac{1}{\nu}}}))^{1 + \frac{1}{\nu}}}{1 + \frac{1}{\nu}}$$ Define $\gamma = (\alpha^{\frac{1}{1 + \frac{1}{\nu}}})$ and rewrite the utility function in terms of $\gamma$ instead of $\alpha$ $$u(c, n) = \log (c) - \frac{(\gamma \cdot n)^{1 + \frac{1}{\nu}}}{1 + \frac{1}{\nu}}$$

Written this way, I see $\gamma$ and therefore $\alpha$ as serving as a unit shifter. How many seconds of leisure makes you indifferent over one dollar less consumption? Say it were 60 seconds. What unit is $n$ measured in? Should we put in $n=1$ for minutes, $n=60$ for seconds, or $n=\frac{1}{60}$ for hours? $\nu$ can't answer that question because it controls the shape and curvature but not the scale of the utility function with respect to $n$. However, $\gamma$ can. If the "true" measure of $n$ is in hours and we guess wrong and choose seconds then $\gamma$ should be $\frac{1}{3600}$ and conditional on $\nu$ this implies $\alpha$. In calibration the true units of $n$ it need not map cleanly to one of our time units. $n$ might be properly measured in units of 15.5 seconds or $\pi$ minutes, but the functional form is flexible enough to handle these cases regardless of if you input the labor in the calibration in any of the more standard units of hours, minutes, or seconds.

As for how to calibrate this, you have to see the bundles of labor/ leisure and consumption chosen by households for given sets of prices. This, combined with the demand functions that follow from this utility setup, should give you the required calibration. The paper A Model of Housing in the Presence of Adjustment Costs: A Structural Interpretation of Habit Persistence (Flavin and Nakagawa (2008), free copy here) shows the way to do this in a different (housing / non-housing consumption instead of consumption / leisure) problem. Their $\gamma$ parameter acts much like $\alpha$ does here.

In Yongsung Chang, Sun-Bin Kim, Frank Schorfheide (2010), NBER paper

the authors use the exact same utility function and they write (p.9)

"Given all other parameters, we set the preference parameter B (i.e. your $\alpha$) such that the steady-state employment rate is 60%, the average employment in our sample period."

In Table 1 of the paper (p.34), we see that this means $\alpha = 101$.